Simulations of infectious diseases on networks

doi:10.1016/j.compbiomed.2005.12.002

Computers in Biology and Medicine

Volume 37, Issue 2, February 2007, Pages 195-205

https://doi.org/10.1016/j.compbiomed.2005.12.002 Get rights and content

Abstract

This paper examines the spread of diseases within populations in the context of networks of potentially disease-causing contacts. We examine the assumptions underlying classical mathematical models of epidemics and how more realistic assumptions can be made using contact networks. Several well-known kinds of contact networks are examined and simulated by evaluating their structural properties relevant to disease propagation. Algorithms used in the study of these networks are explained and numerical simulations of percolation and the epidemic process carried out to explore the effects that the network structure has on disease progression.

Introduction

The purpose of modelling epidemics is to understand the processes by which they spread, and thus provide a rational basis for formulating more effective prevention programmes and combating outbreaks. To be able to decide beforehand whether a targeted vaccination programme is likely to work and to decide which individuals will contribute most to the transmission of the infection, is to make that much more effective use of limited resources in combatting disease spread.

The epidemic process is essentially a population growth model, the disease being represented by infected individuals and the remaining (limited) resources by those susceptible to infection. Models of epidemics are classified by their assumptions about the disease and population [1]. Disease assumptions include the mechanism of infection (such as by direct contact, carriers and vectors), and removal or recovery (fixed time, probabilistic). A model makes assumptions about the population structure, which may be homogeneous except for disease status, consist of heterogeneous subgroups where there are different risk factors (such as age, gender, culture), or be generally heterogeneous on an individual basis, which can be included as a network representation. Regarding population dynamics, the model may assume a closed population, or an open one in which it may grow or shrink over time by including birth and deaths [2].

The primary assumption a model must make is in its choice of an exhaustive and mutually exclusive classification of disease status. The simplest possible model is the SI model in which individuals are either susceptible (S) or infective (I) and the progress of the epidemic is traced by transmission between infectives and susceptibles until it ends (at least mathematically) when the entire population is infected. The model explored here is the SIR model in which infectives become removed (R) probabilistically or after a time period. The R status may correspond to death, quarantine, or recovery with permanent immunity.

There are many variations of these models, such as the SEIR model, which includes an exposed (E) class in which the individual has the disease but does not pass it on to susceptibles. A long incubation period can cause new infectives to arrive in waves [3]. The SIS model includes infectives that become susceptible again, and the SIRS model represents diseases conferring temporary immunity—these models are often used to model different epidemics such as HIV or SARS [2].

In the following sections, we first discuss the traditional fully mixed models and their success at describing the basic features of epidemics, and then how the contact network does away with the assumption of full-mixedness and opens many possibilities for more realistic models. We then investigate the structural measures of networks, the mapping of the final epidemic to bond percolation, and some of the more detailed epidemiological information that can be obtained by the SIR model on networks. We then examine common network models and compare their relative usefulness as models of real-world contact networks in terms of their structural measures. Algorithms for generating the important scale-free network models are described, as well as an efficient means of calculating the mean path length, clustering coefficient and percolation threshold, and carrying out the SIR model on a general network. We finish with a discussion of dynamic networks that do away with the important simplifying assumption of a static network structure and outline the challenges involved in doing so.

Section snippets

Fully mixed models

Traditional models assume that the population is fully mixed, that is, a susceptible has a fixed probability $β$ per unit time of contracting the disease from any infective in the population. Combined with a fixed probability $γ$ per unit time of any infective becoming removed, this allows the number $s, i$ and r of each class of individual in a closed population of size N to be modelled by a system of ordinary differential equations first proposed by Kermack and McKendrick in 1927 [2]: $\frac{d s}{d t} = - β si, \frac{d i}{d t} = β$

Network models

Network models do away with the assumption of a fully-mixed population. A population is represented as a collection of individuals, called vertices or nodes from the terminology of graph theory or sites in the context of percolation. Two nodes are connected by an edge if they are in regular contact and has the potential to transmit the disease if one of the nodes is infective. On a given day it is unlikely that there is a uniform probability of shaking hands with any given person in your city;

Common network models

In this section we describe a number of widely studied networks on which epidemic models are often based, and we provide some analytical properties and common variations. The models described are the random, Watts–Strogatz, lattice, Barabasí–Albert, and power-law or “scale-free” networks, of which all but the lattice exhibit the small world property.

Network algorithms

We now discuss the algorithms implemented here for numerically calculating structural and epidemiological measures on networks.

Dynamic networks

The networks discussed, thus, far have all been accumulated networks of potentially disease-transmitting contacts that occur between individuals on a regular basis. This assumption simplifies computation and is valid possibly on a timescale of several months. However, on longer timescales as in the case of HIV one might consider a regular contact network whose structure changes through addition and deletion of nodes and edges. Unfortunately, the network models here, except the random network,

Conclusions

Using a contact network removes the assumption of a fully-mixed population inherent in traditional models. There are several common network models, each of which has similarities and differences from those expected of typical human contact networks. One important structure, the community, may be possible to incorporate as an additional step. Information can be gathered from a network's structural properties, the mapping of the removed nodes after an epidemic to a bond percolation problem, and

Gareth Witten is a lecturer in the Department of Mathematics and Applied Mathematics at the University of Cape Town. His research is in the applications of quantitative methods to biology and medicine. He leads a vibrant research group under the South African Centre for Epidemiological Modelling and Analysis (www.sacema.ac.za) that focuses on modelling diseases, in particular HIV, within the host. He is also a fellow of the Stellenbosch Institute for Advanced Study, a think-tank for complex

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Graham Poulter is in 2006 beginning research towards an M.Sc. in Computational Biology at the University of Cape Town. He was awarded a B.Sc.Hons with distinction in Applied Mathematics in 2005, and a B.Sc. with distinction in Physics, Applied Mathematics and Computer Science in 2004. The core research for this paper, implementing and reviewing network models of epidemics, was carried out in 2004 during his B.Sc. final year project. E-mail:[email protected], URL:http://mancala.cbio.uct.ac.za/∼graham/

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Simulations of infectious diseases on networks

Abstract

Introduction

Section snippets

Fully mixed models

Network models

Common network models

Network algorithms

Dynamic networks

Conclusions

Math. Biosci.

Epidemic Modelling

The mathematics of infectious diseases

SIAM Rev.

Sexually transmitted diseases—epidemic cycling and immunity

Nature

Collective dynamics of small world networks

Nature

The structure and function of complex networks

SIAM Rev.

Spread of epidemic disease on networks

Phys. Rev. E